On 25 Jan., 08:41, William Hughes <wpihug...@gmail.com> wrote:> On Jan 25, 8:32 am, WM <mueck...@rz.fh-augsburg.de> wrote:>>>>>> > On 25 Jan., 01:27, William Hughes <wpihug...@gmail.com> wrote:>> > > On Jan 24, 8:52 am, WM <mueck...@rz.fh-augsburg.de> wrote:>> > > > The following is copied from Mathematics StackExchange and> > > > MathOverflow. Small wonder that the sources have been deleted already.>> > > > How can we distinguish between that infinite Binary Tree that contains> > > > only all finite initial segments of the infinite paths and that> > > > complete infinite Binary Tree that in addition also contains all> > > > infinite paths?>> > > > Let k denote the L_k th level of the Binary Tree. The set of all> > > > nodes of the Binary Tree defined by the union of all finite initial> > > > segments (L_1, L_2, ..., L_k) of the sequence of levels U{0 ... oo}> > > > (L_1, L_2, ..., L_k) contains (as subsets) all finite initial segments> > > > of all infinite paths. Does it contain (as subsets) the infinite paths> > > > too?>> > > > How could both Binary Trees be distinguished by levels or by nodes?>> > > They cannot of course. Both have exactly the same levels and the same> > > nodes.>> > > They can of course be distinguished.>> > > In one case you do not include infinite subsets.> > > In the other you do.>> > My question aimed at the posiibility to distinguish the Binary Trees> > by a mathematical criterion, namely that one that is applied in the> > diagonal argument. Of course you have understood that.>> > That does not hinder you to believe in addition in matheological> > concepts that cannot be based on mathematical facts like nodes,> > levels, or digits.>> Nope. The concept is based on nodes, and levels.>> We can use the same set of nodes to make two collections of> sets of nodes. One collection contains all sets of nodes, X, with> the property that there is a node in X with a level greater or> equal to that of any other node in X.> The other collection contains all sets of nodes, Y, with the property> that there is no node in Y with a level greater or equal to that of> any other node Y.-

And both sets of nodes are completely exhausted by the same paths withthe only difference that one kind is called X and is finite and theother kind is called Y and is infinite. And, of course, there is abijection between both kinds because they contain same nodes. And,yes, there is a last minor difference, there are far more Y's thanX's. But nobody would notice.